The Double Root

What does the one-solution case look like up close? x^2 - 4x + 4 = 0: the left side passes the perfect-square check, (x - 2)^2 = 0. A square is zero only when its base is: x - 2 = 0, so x = 2, once.

Through the zero product lens the factors are (x - 2)(x - 2): both cases say the same thing. The solution 2 is called a double root: one value doing the work of two.

The discriminant agrees, as it must: 16 - 16 = 0. Zero under the root, ± of nothing, one answer. Three viewpoints, square, factors, discriminant, and they always tell one story.